Abelian fundamental groups

One of the most spectacular consequences of noncommutativity of the group Q is the possibility of a topological entanglement of the disclinations. Originally, the problem was considered by Toulouse for biaxial nematics [42], but it apphes to any medium with a non-Abelian fundamental group [50]-[52. ... 

The fundamental grows of reaction mechanisms are not in general commutative. The fundamental group 111 of level set F(A) is non-Abelian whenever m>l, since the reaction mechanisms [Pi][P2] and [P2HP1] expressed in terms of two generator mechanisms [Pi] and[P2] are two different reduced products in a free group ... 

Consequently, the products [Pi][P2] and [P2][PlJ represent two different reaction mechanisms, hence the group is not commutative. However, every nontrivial fundamental group III always contains a special Abelian subgroup, the infinite cyclic group. Of course, in the case of m=0 the group is also commutative, since then... 

Several generalizations of the above concepts are familiar in particle physics. The assumption that the proton and the neutron are two diflferent states of the same particle, the nucleon, which transform into each other under isotopic spin rotations leads to invariance under the group 517(2) which is the simplest example of a non-Abelian group, i.e. one in which not all transformations commute with each other. In the eight-fold way p, n, A, S" ", S , S°, E are all taken as states of a fundamental baryon and... 

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